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LLL(..., ChangeMatrix => ...) -- also find change of basis matrix

Synopsis

Description

Constructs the change of basis matrix U from the basis A to the basis B. This is an invertible matrix U such thatAU = B.
i1 : A = matrix randomMutableMatrix(10,10,.5,100000)

o1 = | 0      0      -46955 77140  74939  19764  0      -28422 0      0     |
     | 0      0      0      0      0      90542  -16170 -76479 -37214 99345 |
     | -52080 0      96364  -50151 -17453 -65353 0      0      0      0     |
     | -19887 0      0      -46034 -21491 86441  0      -81167 28640  0     |
     | -74491 -76816 0      0      0      45227  66039  79524  0      28859 |
     | -8479  40925  49444  0      -6193  0      14787  0      -17306 39804 |
     | 0      0      -12656 56445  64711  0      -26644 0      -40859 0     |
     | 0      0      0      0      82500  35472  -44479 -99655 0      0     |
     | 56498  -65212 0      85954  72800  30887  0      77187  0      60794 |
     | 0      -49187 47009  -6047  0      0      6041   7632   0      0     |

              10        10
o1 : Matrix ZZ   <--- ZZ
i2 : (B,U) = LLL(A, ChangeMatrix=>true)

o2 = (| -2201  0      2201   2201   -2201  -10859 27191  -28422 52150 10859 
      | -16170 -37214 16170  -4874  4874   14063  -12241 -19222 7367  48068 
      | -19382 0      -32698 19382  -19382 -32655 21069  52080  242   32655 
      | 4656   28640  -24543 23984  -23984 29817  -37914 -4000  35751 -1177 
      | -8452  0      -66039 -57587 -19229 47935  -7916  -26020 24820 -19076
      | 115    -17306 -8594  -32208 73133  34732  -33254 25022  9525  -12234
      | -18378 -40859 18378  4163   -4163  8266   -1559  -28430 62167 -49125
      | 38021  0      -38021 6458   -6458  18317  9007   -10697 -2549 -18317
      | 43344  0      13154  -43344 -21868 29708  -26591 16271  12703 31086 
      | 12088  0      -12088 -18129 -31058 -35508 -53050 -53637 28874 35508 
     ------------------------------------------------------------------------
     |, | 1  0 0  -1 1  0  -1 -1 -1 0  |)
     |  | 0  0 0  0  1  1  0  1  0  -1 |
     |  | 0  0 0  0  0  0  -1 0  1  0  |
     |  | -1 0 1  1  -1 -1 0  0  2  1  |
     |  | 1  0 -1 -1 1  1  0  0  -1 -1 |
     |  | 0  0 0  0  0  1  -1 0  1  -1 |
     |  | 1  0 -1 -2 2  0  -1 -2 -1 0  |
     |  | 0  0 0  0  0  1  0  1  0  -1 |
     |  | 0  1 0  1  -1 0  1  2  0  1  |
     |  | 0  0 0  0  0  0  1  1  -1 1  |

o2 : Sequence
i3 : B == A*U

o3 = true

Further information

See also